I run a lot of 1-1 Dungeons & Dragons games. We pretty much play every day and the vast majority of the games I run are just for my Daddy. His character versus my dungeon or monsters. In the four years I have been running games, I have learned a lot about how to make it fun. I used to have a lot of problems focusing on my own characters or making the plot impossible because I didn't want my cool monster to die, but these days we jump right into the action and the focus in on the player character. our 1-1 games are a blast. Hopefully these tips will help you run great 1-1 games as well.
1. Don't Waste Time: Jump right into the action. There is no need to have character introductions or descriptions. Don't waste time setting things up. Your first words as a DM should be describing the first encounter. Then you can let that encounter lead to what is going on next.
2. Be Prepared for Fast Combat: In a 1-1 game, combat is fast. Characters use up spell slots and abilities quickly and hit points drain in no time. Give the character plenty of warnings if things look overwhelming so they can run away (which is easier with just one person). The player will take an action and in no time it will be their turn again. They will burn through actions that are limited such as smites or lay on hands or rages. Rounds will fly by which could impact spells that last for a minute or abilities that normally have a duration long enough to last for the entire combat. This can be a whole lot of fun and very exciting but it does take some getting used to.
3. Customize the Plot: The plot should revolve around the one character. Give them dreams or use prophesies that focus on them alone. Customize magic items for them. Focus the story on their specific abilities such as a heist for a rogue or a magic test for a wizard or a test of faith for a cleric. You do not have to worry about spreading the plot around between various characters so you can have fun and make everything about the one character. The downside to this is that you pretty much have to customize everything. Prewritten adventures don't really work for 1-1 games except to serve as inspiration. The prewritten adventures often require skills of abilities your one character doesn't have or it might be written requiring more combat power than your character can provide.
4. No DMPCs: NPCs are fine, but if you focus all the attention on your epic character, the player will get bored. One of the coolest parts of 1-1 games is how the PC gets to be the star of every part of the story. Running a DMPC ruins that. Keep the spotlight on the PC. This doesn't mean there shouldn't be other characters in the game, but make them characters that fill a very specific role. See supporting characters below.
5. Supporting Characters: The character will have a very limited set of abilities and will need to have access to other skills such as healing or magic item identification. Create a home base that the character can return to to get these things. Maybe they have some friends that have some great abilities but just don't want to go on the main adventure. Maybe they have access to a temple for healing and potions. Maybe they have a patron that provides them with access to things they can't do themselves. You can also use hirelings or henchmen, but just don't kmake those people the star of the adventure.
6. Make it Dangerous: One thing people complain about in 5e is that it isn't deadly enough. Well 1-1 games can be very deadly. A few bad rolls and the character is unconscious with no one to help them. No healing word to wake them up. No spare the dying. Embrace the danger, and make things scary. Give the character plenty of chances to run away and come back to fight another day. Make sure they are aware of how bad things could get if they are unconscious or dying. That being said, I tend to avoid spells that stun or incapacitate the character. Those aren't much fun in a normal party, but they can ruin an encounter with only a single player.
7. Death: I normally avoid killing characters in 1-1 games. If the PC falls unconscious, have them wake up as a prisoner of the bad guy or wrapped up in spiderwebs or being carried back to the monster's babies so they can feed on something that is alive. Death is boring.
8. Death Again: In 1-1 games, you can easily explore what happens when a character dies. If you want to include death (it isn't always boring) you can have a character wake up in heaven or hell or the outer planes or the spirit world. What happens when they die? This is tough in a party with multiple players unless they al die at the same time. If a single character dies and you want to explore the afterlife, you have just split the party and that never goes well. In 1-1 games, you have every opportunity to explore this fascinating part of your world.
9. Unique Scenarios: In 1-1 games, you can explore unique scenarios that you could never run with a large party. The spy who for a month has been impersonating a king's guard is really boring for the majority of the party, but great for a 1-1 game. Quests to find spell components or magic sword oil are pretty specific to one player which make them great for 1-1 games. Exploring family relations or delving into a backstory are so easy when you only have one player.
10. Explore Who They Are: Players get to really explore who their characters are in 1-1 games because everything revolves around them. They are constantly figuring out what they are going to do and why. They can become experts on their characters motivations and abilities. They can truly develop their backstories and personal goals.
